The 1917 LPs give HRH to the following people:
1. The children of the monarch* (Andrew and Edward)
2. The spouses of the sons of the monarch - (Camilla, Sophie)
3. The male-line grandchildren of the monarch* (William, Harry, Beatrice, Eugenie, Richard, Edward, Michael and Alexandra - Richard, Michael, Alexandra and the late William of Gloucester all were born after the death of George V but still they were born as HRHs)
4. The spouses of male-line grandsons of the monarch (Kate, Birgitte, Katherine, Marie-Christine)
5. The eldest son of the eldest son of the Prince of Wales - George*
*1 - Charles and Anne were HRHs from birth because George VI issued special LPs in 1948 to give that status to the children of Elizabeth. Without those special LPs then Charles and Anne would have been born as Lord Charles Mountbatten, Earl of Merioneth and Lady Anne Mountbatten. They would have been raised to HRH when the Queen ascended the throne in 1952 without those special LPs from 1948
*3 - Louise and James aren't HRHs as the Queen's will was made known that they would be styled as the children of an Earl. There are some people who argue that only an LP can overturn an LP but that actually isn't the case e.g. William was called Duke of Cambridge from his wedding day but the LPs weren't issued for another few weeks - in May not April. The Queen's Will is all that is needed (I wrote to BP on this question and that was the answer - the Queen's Will is all that was needed to say that they aren't Prince or Princess and that is good enough for me)
*5 - In 2012 The Queen issued to LPs to extend that to all the children rather than just the eldest son. That was necessary as they law about making birth order alone rather than gender bias meant a first born girl would be the future monarch. Under the 1917 LPs a first born girl would have been Lady xxxx Mountbatten-Windsor while a later son would have been born HRH but the Lady would have been the future monarch.
With that understanding of the 1917 LPs currently Harry's children will be born as Lord/Lady Mountbatten-Windsor and be immediately promoted to HRHs when Charles becomes King. If a child is born after Charles accession that child automatically be HRH, in much the same was as Andrew and Edward were born as HRHs under the 1917 LPs while Charles and Anne needed the 1948 LPs.
Next - what would happen to Harry's children if Charles predeceases the Queen and no special LPs have already been issued. Then, under the existing LPs, they remains as Lord/Lady. William could then issue new LPs on the basis that they would get HRH on the accession of their grandfather. Whether he would do so would depend on the circumstances at the time.